Variational approach to the Coulomb problem on a cylinder

We evaluate, by means of variational calculations, the bound state energy EB of a pair of charges located on the surface of a cylinder, interacting via Coulomb potential - e²/^r. The trial wave function involves three variational parameters. EB is obtained as a function of the reduced curvature C=a₀/R, where a₀ is the Bohr radius and R is the radius of the cylinder. We find that the energetics of binding exhibits a monotonic trend as a function of C; the known one- and two-dimensional limits of EB are reproduced accurately by our calculation. E_B is relatively insensitive to curvature for small C. Its value is ∼1% higher at C = 1 than at C = 0. This weak dependence is confirmed by a perturbation theory calculation. The high curvature regime approximates the one-dimensional Coulomb model; within our variational approach, E_B has a logarithmic divergence as R approaches zero. The proposed variational method is applied to the case of donors in single-wall carbon nanotubes.